Difficult theoretical problem on basis vectors

[spaghetti3451]

How the hell do you prove that the components of a vector w.r.t. a given basis are unique?

I have literally no idea how to begin! It's just that with these theoretical problems there's no straightforward starting point!

 

[HallsofIvy]

Suppose they were not unique. That is, suppose there were some vector, v, such that
$$v= a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_nv_n$$
and
$$v= b_1v_1+ b_2v_2+ \cdot\cdot\cdot+ b_nv_n$$
Now subtract:
$$(a_1- b_1)v_1+ (a_2- b_2)v_2+ \cdot\cdot\cdot+ (a_n- b_n)v_n= 0$$

One of the requirements for a "basis" is that the vectors are independent. Use the definition of "independent".

 

[spaghetti3451]

I see! If the basis vectors are independent, then the coefficients are zero, that is, the components are unique.

I have to tell you, though! It's not easy to figure out how to go about the problem.

I just wish the solution were as obvious as it looks now!

 

[dodo]

If it helps, the general pattern of any proof about "uniqueness" is to present two elements, and deduce that they are the same. The first two lines of Halls' proof are exactly that... and starting is usually the hardest thing. Once into it, you'll figure out a way of proving that your two elements are equal.